Question

Find the derivative of $ \left| {2{x^2} - 3} \right| $ with respect to $ x $ .

Answer 1

Hint: The sum rule and constant rule is applicable here. We need to differentiate the variable $ 2{x^2} $ and $ 3 $ differently by using different rules applicable to each of them.

Formula used: The formulae used in the solution are given here.
 $ \dfrac{d}{{dx}}n{x^a} = na{x^{a - 1}} $ where $ a $ and $ n $ are real numbers.
By sum rule, $ \left( {\alpha f + \beta g} \right)' = \alpha f' + \beta g' $ for all functions $ f $ and $ g $ and for all real numbers $ \alpha $ and $ \beta $ .
By constant rule, if $ f\left( x \right) $ is a constant, then its derivative is $ f'\left( x \right) = 0. $

Complete Step by Step Solution
The function given here is $ f\left( x \right) = \left| {2{x^2} - 3} \right| $ .
Differentiation is the action of computing a derivative. The derivative of a function $ y = f\left( x \right) $ of a variable $ x $ is a measure of the rate at which the value $ y $ of the function changes with respect to the change of the variable $ x $ . It is called the derivative of $ f $ with respect to $ x $ .
If $ x $ and $ y $ are real numbers, and if the graph of $ f $ is plotted against $ x $ , the derivative is the slope of this graph at each point.
To find the derivative, we differentiate $ f\left( x \right) $ with respect to $ x $ .
 $ f'\left( x \right) = \dfrac{{d\left( {f\left( x \right)} \right)}}{{dx}} = \dfrac{d}{{dx}}\left( {2{x^2} - 3} \right) $
According to the sum rule, which states that, $ \left( {\alpha f + \beta g} \right)' = \alpha f' + \beta g' $ for all functions $ f $ and $ g $ and for all real numbers $ \alpha $ and $ \beta $ .
We can write this in a simpler way as, $ \dfrac{d}{{dx}}\left( {2{x^2}} \right) - \dfrac{d}{{dx}}\left( 3 \right) = 4x $ .
This is according to, $ \dfrac{d}{{dx}}n{x^a} = na{x^{a - 1}} $ where $ a $ and $ n $ are real numbers.
By constant rule, if $ f\left( x \right) $ is a constant, then its derivative is $ f'\left( x \right) = 0. $
Thus, the value of $ \dfrac{d}{{dx}}\left( 3 \right) $ is zero, since $ 3 $ is a constant.

Note
Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century. However, each inventor claimed the other stole his work in a bitter dispute that continued until the end of their lives.